The effect of strongly temperature-dependent viscosity on convection in the interior of Venus is studied systematically with the help of finite element numerical models. For viscosity contrasts satisfying experimental constraints on the rheology of rocks, Venus is likely to be in the regime of stagnant lid convection. This regime is characterized by the formation of a slowly creeping, very viscous lid on top of the mantle-Venusian lithosphere and is in agreement with the tectonic style observed on Venus. Stagnant lid convection explains large geoid to topography ratios on Venus by the thermal thinning of a thick lithosphere. The thickness of the lithosphere can be as large as 400--500 km for Beta Regio and 200--400 km on average. Geoid and topography data and experimental data on the rheology of rocks provide constraints on the viscosity of the mantle, 1020--1021 Pa s; the convective stresses in the interior, 0.2--0.5 MPa; the stresses in the lid, 100--200 MPa; the velocity in the interior, 0.5--3 cm yr-1; and the heat flux beneath the lithosphere, 8--16 mW m-2. Parameterized convection calculations of thermal history of Venus are difficult to reconcile with a thick present-day lithosphere. However, a sufficiently thick lithosphere can be formed if a convective regime with mobile plates was replaced by stagnant lid convection around 0.5 b.y. ago. One of the possible explanations for the cessation of Venusian plate tectonics is that during the evolution of Venus, stresses in the lid dropped below the yield strength of the lithosphere. This model predicts a drastic drop in the heat flux, thickening of the lithosphere, and suppression of melting and is consistent with the hypothesis of cessation of resurfacing on Venus around 0.5 b.y. ago. ¿ American Geophysical Union 1996